Assume that a quality characteristic is normally distributed. An x-bar control chart is constructed with 3 sigma control limits. A rule to decide that the process is out of control is given by the following. If all of the next 7 points fall on the same side of the center line, conclude that the process is out of control.
A) For this rule, find the probability that the process will be found out of control when in fact it is in control.
B) Suppose that the x-bar chart is based on samples of size 1. Further, suppose the mean of the quality characteristic shifts one standard deviation, that is u = u0 + sigma, where u0 denotes the in-control mean (i.e., u0 is the center of the chart), and stays there during the collection of the next 7 samples. What is the probability that the process will be found in control?